A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Author

Reinhard Zumkeller, Nov 05 2001

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)
• Index entries for characteristic functions
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Characteristic function of A028260 (positions of 1's). Cf. also A026424 (positions of 0's) and A320655.

One less than A007421.

Cf. A003961, A008836, A010052, A038548 (inverse Möbius transform), A046523, A055037 (partial sums), A343784, A347102, A353337, A353338, A353555, A353557, A353629, A353669, A358750, A358752, A353374, A358775, A356163, A356170.

Cf. also A066829, A353374.

Programs

Maple

A065043 := proc(n)
    if type(numtheory[bigomega](n),'even') then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 26 2013

Mathematica

Table[(LiouvilleLambda[n]+1)/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)

PARI

{ for (n=1, 1000, a=1 - bigomega(n)%2; write("b065043.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

PARI

A065043(n) = (1 - (bigomega(n)%2)); \\ Antti Karttunen, Apr 19 2022

Python

from operator import ixor
from functools import reduce
from sympy import factorint
def A065043(n): return (reduce(ixor, factorint(n).values(),0)&1)^1 # Chai Wah Wu, Jan 01 2023

Formula

a(n) = 1 - A001222(n) mod 2.

a(n) = A007421(n) - 1.

a(n) = 1 - A066829(n).

a(A028260(k)) = 1 and a(A026424(k)) = 0 for all k.

Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012

a(n) = (A008836(n) + 1)/2. - Enrique Pérez Herrero, Jul 07 2012

a(n) = A001222(2n) mod 2. - Wesley Ivan Hurt, Jun 22 2013

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^2)/(1 - x^n). - Ilya Gutkovskiy, Apr 25 2017

From Antti Karttunen, Dec 01 2022: (Start)

For x, y >= 1, a(x*y) = 1 - abs(a(x)-a(y)).

a(n) = a(A046523(n)) = A356163(A003961(n)).

a(n) = A000035(A356163(n)+A347102(n)).

a(n) = A010052(n) + A353669(n).

a(n) = A353555(n) + A353557(n).

a(n) = A358750(n) + A358752(n).

a(n) = A353374(n) + A358775(n).

a(n) >= A356170(n).

(End)

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009